Abstract
A system of setsE 1,E 2, ...,E k ⊂X is said to be disjointly representable if there existx 1,x 2, ...,x k teX such thatx i teE j ⇔i=j. Letf(r, k) denote the maximal size of anr-uniform set-system containing nok disjointly representable members. In the first section the exact value off(r, 3) is determined and (asymptotically sharp) bounds onf(r, k),k>3 are established. The last two sections contain some generalizations, in particular we prove an analogue of Sauer’ theorem [16] for uniform set-systems.
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Dedicated to Paul Erdős on his seventieth birthday